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## Mathematical Statistics

by: Mrs. Triston Collier

19

0

11

# Mathematical Statistics STAT 710

Mrs. Triston Collier
UW
GPA 3.57

Jun Shao

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COURSE
PROF.
Jun Shao
TYPE
Class Notes
PAGES
11
WORDS
KARMA
25 ?

## Popular in Statistics

This 11 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 710 at University of Wisconsin - Madison taught by Jun Shao in Fall. Since its upload, it has received 19 views. For similar materials see /class/205097/stat-710-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.

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Date Created: 09/17/15
Stat 710 Mathematical Statistics Lecture 14 Jun Shao Department of Statistics University of Wisdonsin Madison WI 53706 USA Liaam M Q7 SQoriri39gand RLE The method of estimating 9 by solving 5y 0 over 9 e e is called scoring and the function 5y is called the score function RLE s are not necessarily MLE s We may use the techniques discussed in 44 to check whether an RLE is an MLE However according to Theorem 417 when a sequence of RLE s is consistent then it is asymptotically efficient We may not need to search for MLE s if asymptotic efficiency is the only criterion to select estimators Typically a sequence of MLE s is consistenct and asymptotically efficient although there are examples in which an RLE sequence is consistent but an MLE sequence is not Example 439 Suppose that X has a distribution in a natural exponential family ie the pdf of X is fnXi eXPMTTOQ C71hxi Since leogfnx8718nT 782 nBn8n7 condition 82 log fyx SU BYBYT v HY THKCU hm is satisfied From Proposition 32 other conditions in Theorem 416 are also satisfied For iid X s Example 439 continued If E n 1ZL1TXe e the range of 9 gn 8 nBn then E is a unique RLE of 9 which is also a unique MLE of 9 since yam8718717 VarTX is positive definite Also 71 g 19 exists and a unique RLE MLE of n is n g 1 n However 3 may not be in e and the previous argument fails eg Example 429 What Theorem 417 tells us in this case is that as n a co P 7 E e 1 and therefore E or n is the unique asymptotically efficient RLE MLE of 9 or n in the limiting sense In an example like this we may directly show that P37 e e a 1 using the fact that 3 Ha ETX1 901 the SLLN Example 439 continued If E n 1ZL1TXe e the range of 9 gn 8 nBn then E is a unique RLE of 9 which is also a unique MLE of 9 since yam8718717 VarTX is positive definite Also 71 g 19 exists and a unique RLE MLE of n is n g 1 n However 3 may not be in e and the previous argument fails eg Example 429 What Theorem 417 tells us in this case is that as n a co P 7 E e 1 and therefore E or n is the unique asymptotically efficient RLE MLE of 9 or n in the limiting sense In an example like this we may directly show that P37 e e a 1 using the fact that 3 Ha ETX1 901 the SLLN The next theorem provides a similar result for the MLE or RLE in the GLM 442 lts proof is similar to the proof of Theorem 417 Consider the GLM with d t and t s in a fixed interval t0tw O lt to too lt co Assume that the range of the unknown parameter 3 is an open subset of 3 at the true value of B O lt infpBTZ g sup pBTZ lt m where pt w t2 vt as n a co maxign ZTZTZ 1Z a O and 7LZTZ a co where Z is the nx p matrix whose ith row is the vector Z and 7LA is the smallest eigenvalue ofA i There is a unique sequence of estimators 3 such that Ps 0 H1 and En Hp 37 where 513 Blog BB is the score function ii Let IB VarsB Then ln 12Eni ed Npol iii If d is known orthe pdf indexed by 9 B satisfies the conditions for f9 in Theorem 416 then 7 is as mtoticall efficient Assume the conditions in Theorem 416 Let 5y be the score function Let 330 be an estimator of 9 that may not be asymptotically efficient The estimator an 95Vsne r1sne is the first iteration in computing an MLE or RLE using the NewtonRaphson iteration method with 330 as the initial value and therefore is called the onestep MLE Without any further iteration 331 is asymptotically efficient under some conditions OneStep MLE Assume the conditions in Theorem 416 Let 5y be the score function Let 330 be an estimator of 9 that may not be asymptotically efficient The estimator n 30eVsn r1sn is the first iteration in computing an MLE or RLE using the NewtonRaphson iteration method with 330 as the initial value and therefore is called the onestep MLE Without any further iteration 331 is asymptotically efficient under some conditions Theorem 419 Assume that the conditions in Theorem 416 hold and that 330 is consistent for 9 Definition 210 i The onestep MLE 331 is asymptotically efficient ii The onestep MLE obtained by replacing V5y with its expected value Iy the Fisherscoring method is asymptotically efficient Since ago is consistent we can focus on the event 330 6 Ag yr iii79H S s for a sufficiently small 8 such that Ag C 9 From the meanvalue theorem 5n 7 sne M1Vsnet 7 9dt 330 7 9 Substituting this into the formular for 331 we obtain that 63 7 e ewsnr nwhrisnw ilk e Gn i t 9 where Gn 7 Vsn go 101Vsn9 630 7 9dt From the proof of Theorem 417 HInei12Vsn r1 In912 IkH Hp o Proof continued Using an argument similar to those in the proof of Theorem 417 we can show that llGn r20kll Hp 0 These results and the fact that Jag 7 9 Op1 imply mote iln9r1sneop1 This proves i The proof for ii is similar

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